Geodesic triangle

A figure consisting of three different points together with the pairwise-connecting geodesic lines (cf. Geodesic line). The points are known as the vertices, while the geodesic lines are known as the sides of the triangle. A geodesic triangle can be considered in any space in which geodesics exist.

If the sides of a geodesic triangle situated in a region homeomorphic to an open disc constitute a simply-closed contour, then the interior domain is added to the geodesic triangle. On a regular surface the sum of the angles of a geodesic triangle minus $\pi$ (the excess of the triangle) is equal to the total curvature of the interior region [1].

Given a geodesic triangle in a metric space, a plane triangle with the same side lengths is often considered. This makes it possible to introduce various concepts of an angle between two shortest lines in metric spaces. In the two-dimensional case, after an angle measurement has been introduced, it is possible to introduce the total curvature as a set function expressed in terms of the excess of geodesic triangles. Nets of geodesic triangles serve as a source of the approximation of metrics by the polyhedral metrics [2].

Estimates are available of the difference between the angle of a geodesic triangle in the space under consideration and the respective angle in a triangle with the same side lengths in a plane or on a surface of constant curvature [1], [3], [4].

References

[1]	C.F. Gauss, "Allgemeine Flächentheorie" , W. Engelmann , Leipzig (1900) (Translated from Latin)
[2]	A.D. Aleksandrov, V.A. Zalgaller, "Intrinsic geometry of surfaces" , Transl. Math. Monogr. , Amer. Math. Soc. (1967) (Translated from Russian)
[3]	A.D. Aleksandrov, "A theorem on triangles in a metric space and some of its applications" Trudy Mat. Inst. Steklov , 38 (1951) pp. 5–23 (In Russian)
[4]	D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968)

References

[a1]	W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German)
[a2]	J. Cheeger, D.G. Ebin, "Comparison theorems in Riemannian geometry" , North-Holland (1975)
[a3]	J. Cheeger, W. Müller, R. Schrader, "On the curvature of piecewise flat spaces" Comm. Math. Physics , 92 (1984) pp. 405–454